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Jun 27, 2018
06/18

by
Boris Haspot; Ewelina Zatorska

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We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\frac{1}{\sqrt{\varepsilon}}$ for $\varepsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation-$\rho_\varepsilon$ converges to the unique solution to the porous medium equation...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1504.04219

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Jun 30, 2018
06/18

by
Charlotte Perrin; Ewelina Zatorska

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We approximate a two--phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by P.L. Lions and N. Masmoudi [ Annales I.H.P., 1999]. The paper is an extension of the previous result obtained in one-dimensional setting by D. Bresch et al. [ C. R. Acad. Sciences Paris, 2014] to the multi-dimensional case...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1410.0498

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Jun 29, 2018
06/18

by
Pierre Degond; Piotr Minakowski; Ewelina Zatorska

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We study the existence of weak solutions to the two-phase model of crowd motion. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime $\varrho^*(t,x)$ achieves a threshold value $\varrho^*(t,x)$ that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow....

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1612.08411

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Jun 30, 2018
06/18

by
Didier Bresch; Benoit Desjardins; Ewelina Zatorska

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This paper addresses the issue of global existence of so-called $\kappa$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $\kappa$-entropy. This new entropy involves a mixture parameter...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1411.5488

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Jun 29, 2018
06/18

by
Julien Barré; Pierre Degond; Ewelina Zatorska

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We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1607.01975

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Jun 28, 2018
06/18

by
Piotr B. Mucha; Milan Pokorný; Ewelina Zatorska

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The steady compressible Navier--Stokes--Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e. the adiabatic constant $\gamma$ appearing in the pressure law $p(\vr,\vt) \sim \vr^\gamma + \vr \vt$ and the growth exponent in the heat conductivity, i.e. $\kappa(\vt) \sim (1+ \vt^m)$, and without any restriction...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1511.06521

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Jun 30, 2018
06/18

by
Didier Bresch; Vincent Giovangigli; Ewelina Zatorska

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In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1411.5482

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Jun 30, 2018
06/18

by
Piotr Bogsław Mucha; Milan Pokorný; Ewelina Zatorska

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We investigate a coupling between the compressible Navier-Stokes-Fourier system and the full Maxwell-Stefan equations. This model describes the motion of chemically reacting heat-conducting gaseous mixture. The viscosity coefficients are density-dependent functions vanishing on vacuum and the internal pressure depends on species concentrations. By several levels of approximation we prove the global-in-time existence of weak solutions on the three-dimensional torus.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1401.5112

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Jun 28, 2018
06/18

by
Eduard Feireisl; Rupert Klein; Antonin Novotny; Ewelina Zatorska

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We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable -- the potential temperature $\Theta$. We consider the case of well-prepared initial data on "flat" tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1506.06916

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Jun 29, 2018
06/18

by
José A. Carrillo; Young-Pil Choi; Ewelina Zatorska

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We analyse the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.05229

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Jun 29, 2018
06/18

by
David Maltese; Martin Michalek; Piotr B. Mucha; Antonin Novotny; Milan Pokorny; Ewelina Zatorska

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We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the Navier-Stokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1603.08965

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Jun 30, 2018
06/18

by
Julien Barré; José Antonio Carrillo de la Plata; Pierre Degond; Diane Peurichard; Ewelina Zatorska

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We provide a numerical study of the macroscopic model of [3] derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodelling process is very fast, the macroscopic model takes the form of a single aggregation diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the 1-dimensional case, we show...

Topics: Statistical Mechanics, Condensed Matter

Source: http://arxiv.org/abs/1701.01435